Feed-forward carrier phase recovery for optical communications

ABSTRACT

The carrier phase of a carrier wave modulated with information symbols is recovered with a multi-stage, feed-forward carrier phase recovery method. A series of digital signals corresponding to the information signals is received. For each digital signal, a coarse phase recovery is performed to determine a first phase angle which provides a first best estimate of the information symbol corresponding to the digital signal. Using the first best estimate as input, a second stage of estimation is then performed to determine a second phase angle which provides an improved (second) best estimate of the information symbol. Additional stages of estimation can be performed. The multi-stage, feed-forward carrier phase recovery method retains the same linewidth tolerance as a single-stage full blind phase search method; however, the required computational power is substantially reduced. The multi-stage, feed-forward carrier phase recovery method is highly efficient for M-QAM optical signals.

This application is a continuation of prior application Ser. No.12/821,426, filed Jun. 23, 2010, which is hereby incorporated byreference.

BACKGROUND

The present disclosure relates generally to optical communications, andmore particularly to feed-forward carrier phase recovery.

The popularity of multimedia communications services over packet datanetworks, such as the Internet, continues to grow; consequently, thedemand for higher capacity in core data transport networks continues togrow. For service providers, core data transport networks are opticalnetworks based on fiberoptic technology. To increase the capacity ofoptical networks, advanced signal modulation techniques, such asquadrature amplitude modulation (QAM) and quadrature phase shift key(QPSK) have been developed. In particular, M-ary QAM (M-QAM) (such assquare 16-QAM and 64-QAM) have the potential to realize opticaltransmission at very high speeds with high spectral efficiency.

Digital coherent detection has proven to be an effective technique fordetecting and demodulating the received optical signals. A key step indigital coherent detection is carrier phase recovery. Carrier phase isdegraded by laser phase noise in the received optical signal. Laserphase noise is dependent on the linewidth of the optical carrier. Forhigh-order M-QAM modulation formats (M>4), the tolerance for laser phasenoise becomes smaller as the modulation level increases, because theEuclidean distance becomes smaller. Consequently, carrier phase recoverymethods with improved laser linewidth tolerance are critical forsuccessful implementation of high-order M-QAM modulation formats.

Various carrier phase recovery methods have been developed. One methodis based on a decision-directed phase-locked loop. This method hasrelatively poor laser linewidth tolerance because the phase estimate isbased on a previous set of data symbols, not the most current datasymbols. In practice, carrier phase recovery methods are implemented inhardware using parallel and pipeline architectures to attain real-timehigh-speed systems. The tolerance for laser linewidth can then becomeworse due to extended feedback delay.

A second carrier phase recovery method is based on the classicfeed-forward M-th power algorithm using dedicated symbols. Because onlya small portion of the symbols can be used for phase estimate forhigh-order M-QAM, however, this method also has inherently poor laserlinewidth tolerance.

A third carrier phase recovery method is based on a blind phase searchalgorithm. Since this method employs a feed-forward configuration andalso involves all the current symbols for phase estimate, it can achievemuch better laser linewidth tolerance than the previous two methods.This method, however, is complex because the required number of testphase angles increases with the modulation level. For high-order M-QAM,the required number is very high; for example, >32 is required forsquare 64-QAM. Since testing a single phase angle by itself requires aseries of computationally intensive steps (rotate a set of data symbols,make a decision, and calculate the mean squared error), thecomputational power required for real-time testing of a large number ofphase angles is very high.

BRIEF SUMMARY

The carrier phase of a carrier wave modulated with information symbolsis recovered with a multi-stage carrier phase recovery method. A seriesof digital signals corresponding to the information signals is received.For each digital signal, an initial coarse phase recovery is performedto determine a first phase angle which provides a first best estimate ofthe information symbol corresponding to the digital signal. Embodimentsof the initial coarse phase recovery include a coarse blind phase searchand a decision-directed phase-locked loop. A second stage of estimationis then performed to determine a second phase angle which provides asecond best estimate of the information symbol. The second best estimateis based at least in part on the first best estimate. Embodiments of thesecond stage of estimation can be a maximum likelihood estimate, anaverage phase rotation estimate, and a restricted blind phase searchestimate.

These and other advantages of the disclosure will be apparent to thoseof ordinary skill in the art by reference to the following detaileddescription and the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic of a generic optical communications system;

FIG. 2 shows a schematic of an optical transmitter;

FIG. 3 shows a schematic of an optical receiver;

FIG. 4A and FIG. 4B show a schematic of a single-stage blind phasesearch method;

FIG. 5A and FIG. 5B show schematics of multi-stage carrier phaserecovery methods;

FIG. 6 shows plots of bit error rate as a function of the logarithm ofthe effective number of test phases;

FIG. 7 shows a flowchart of a multi-stage carrier phase recovery method;and

FIG. 8 shows a schematic of a computational system for implementing amulti-stage carrier phase recovery method.

DETAILED DESCRIPTION

FIG. 1 shows a schematic of a generic optical telecommunications system.Multiple optical transceivers (XCVRs) send and receive lightwave signalsvia optical transport network 102. Shown are four representativetransceivers, referenced as XCVR 1 104, XCVR 2 106, XCVR 3 108, and XCVR4 110, respectively. In some optical telecommunications systems, opticaltransport network 102 can include all optical components. In otheroptical telecommunications systems, optical transport network 102 caninclude a combination of optical and optoelectronic components. Thetransport medium in optical transport network 102 is typically opticalfiber; however, other transport medium (such as air, in the case offree-space optics) can be deployed.

Each transceiver has a corresponding transmit wavelength (λ_(n,T)) and acorresponding receive wavelength (λ_(n,R)), where n=1−4. In some opticaltelecommunications systems, the transmit and receive wavelengths for aspecific transceiver are the same. In other optical telecommunicationssystems, the transmit and receive wavelengths for a specific transceiverare different. In some optical telecommunications systems, the transmitand receive wavelengths for at least two separate transceivers are thesame. In other optical telecommunications systems, the transmit andreceive wavelengths for any two separate transceivers are different.

FIG. 2 shows a schematic of an example of an optical transmitter.Transmit (Tx) laser optical source 202 transmits a continuous wave (CW)optical beam 201 (with wavelength λ) into electro-optical modulator 204,which is driven by electrical signal 203 generated by electrical signalsource 206. Electrical signal 203 consists of an electrical carrier wavemodulated with information symbols (data symbols). The output ofelectro-optical modulator 204 is carrier optical beam 205, whichconsists of a corresponding optical carrier wave modulated withinformation symbols. In general, the amplitude, frequency, and phase ofthe optical carrier wave can be modulated with information symbols.Carrier optical beam 205 is transmitted to optical transport network 102(see FIG. 1).

FIG. 3 shows a schematic of an example of an optical receiver. Carrieroptical beam 301, with wavelength λ, is received from optical transportnetwork 102 (see FIG. 1). Carrier optical beam 301 has an opticalcarrier wave modulated with information symbols. In general, the opticalreceiver determines the amplitude, frequency, and phase of the modulatedoptical carrier wave to recover and decode the information symbols.Carrier optical beam 301 is transmitted into optical coherent mixer 302.Local oscillator laser optical source 304 generates a reference opticalbeam 303, with wavelength λ, modulated with an optical reference wavewith tunable reference amplitudes, reference frequencies, and referencephases. Reference optical beam 303 is transmitted into optical coherentmixer 302.

Optical coherent mixer 302 splits carrier optical beam 301 into carrieroptical beam 301A and carrier optical beam 301B. Optical coherent mixer302 splits reference optical beam 303 into reference optical beam 303Aand reference optical beam 303B, which is phase-shifted by 90 degreesfrom reference optical beam 303A. The four optical beams are transmittedinto optoelectronic converter 306, which contains a pair ofphotodetectors (not shown). One photodetector receives carrier opticalbeam 301A and reference optical beam 303A to generate analog in-phaseelectrical signal 307A. The other photodetector receives carrier opticalbeam 301B and reference optical beam 303B to generate analogquadrature-phase electrical signal 307B. Analog in-phase electricalsignal 307A and analog quadrature-phase electrical signal 307B aretransmitted into analog/digital converter (ADC) 308. The output of ADC308, represented schematically as a single digital stream, digitalsignal 309, is transmitted into digital signal processor 310. Digitalsignal processor 310 performs multiple operations, including timingsynchronization, equalization, carrier frequency recovery, carrier phaserecovery, and decoding.

An optical signal degrades as it propagates from the optical transmitterto the optical receiver. In particular, laser phase noise introducessome uncertainty in the carrier phase of the received signal relative tothe carrier phase of the transmitted signal assuming no laser phasenoise. Carrier phase recovery refers to recovery of the correct carrierphase (carrier phase as originally transmitted assuming no laser phasenoise) from the received signal. In practice, a best estimate of thecarrier phase is determined from the received signal such that a decodedinformation symbol at the receiver is a best estimate of thecorresponding encoded information symbol at the transmitter. Carrierphase recovery determines the phase angle by which an initial decodedinformation signal is rotated to yield the best estimate of thecorresponding encoded information signal.

FIG. 4A and FIG. 4B show schematic block diagrams for a prior-artfeed-forward carrier phase recovery algorithm based on a blind phasesearch method (T. Pfau et al., “Hardware-Efficient Coherent DigitalReceiver Concept With Feedforward Carrier Recovery for M-QAMConstellations,” J. Lightwave Technology, Vol. 27, No. 8, Apr. 15, 2009,pp. 989-999). The blocks refer to functional entities. FIG. 4A shows onetest phase block 402-m, for a test carrier phase angle φ_(m), where

$\begin{matrix}{{\phi_{m} = {\frac{m - 1}{M} \cdot \frac{\pi}{2}}},{m \in {\left\{ {1,2,{\cdots\mspace{14mu} M}} \right\}.}}} & ({E1})\end{matrix}$

The digitized signal (one sample per symbol) entering into the testphase block 402-m is denoted as X_(k) 401. To recover carrier phase in apure feed-forward approach, the received signal X_(k) is inputted intomultiplier operator 422 and multiplied by e^(jφ) ^(m) 423 to rotateX_(k) by test carrier phase angle φ_(m). The output of multiplieroperator 422 is output 425, which is the rotated symbol X_(k)e^(jφ) ^(m). The quantity X_(k)e^(jφ) ^(m) is inputted into decision block 424,which makes the best estimate of the originally transmitted signal basedon the received signal. The output of decision block 424 is output403-m, which is the decoded symbol Ŷ_(k,m). The quantities X_(k)e^(jφ)^(m) and Ŷ_(k,m) are inputted into add/subtract operator 426. The outputof add/subtract operator 426 is output 427, d_(k,m), whered_(k,m)=X_(k)e^(jφ) ^(m) −Ŷ_(k,m). The quantity d_(k,m) is then inputtedinto squared-distance operator 428. The output of squared-distanceoperator 428 is output 429, |d_(k,m)|², which is the squared distance tothe closest constellation point calculated in the complex plane:

$\begin{matrix}{{d_{k,m}}^{2} = {{{{X_{k}{\mathbb{e}}^{{j\phi}_{m}}} - \left\{ {X_{k}{\mathbb{e}}^{{j\phi}_{m}}} \right\}_{D}}}^{2} = {{{{X_{k}{\mathbb{e}}^{{j\phi}_{m}}} - {\hat{Y}}_{k,m}}}^{2}.}}} & ({E2})\end{matrix}$where the subscript D refers to the output of decision block 424.

To remove distortions from additive noise, the squared distances of 2Nconsecutive test symbols rotated by the same carrier phase angle φ_(m)are summed up: the quantity |d_(k,m)|² is inputted into summationoperator 430. The output of summation operator 430 is output 405-m,e_(k,m), where

$\begin{matrix}{e_{k,m} = {\sum\limits_{n = {{- N} + 1}}^{N}{{d_{{k + n},m}}^{2}.}}} & ({E3})\end{matrix}$The optimal value of the filter width 2N depends on the product of thelaser linewidth times the symbol rate.

Refer to FIG. 4B for a schematic block diagram of the overallsingle-stage blind phase search process 400. The operations shown inFIG. 4A are processed in parallel for all

${\phi_{m} = {\frac{m - 1}{M} \cdot \frac{\pi}{2}}},{m \in \left\{ {1,2,{\ldots\mspace{14mu} M}} \right\}},$mε{1, 2, . . . M}, as represented by test phase block 402-1 to testphase block 402-M. The corresponding decoded output symbols, Ŷ_(k,1)403-1 to Ŷ_(k,M) 403-M, are inputted into M×1 switch 404. The sums ofthe squared distances, e_(k,1) 405-1 to e_(k,M) 405-M are inputted intominimization search block 406. The optimum phase angle is determined bysearching for the minimum sum of the squared distances. Thecorresponding decoded output symbol Ŷ_(k) 407 can be selected from theset of values Ŷ_(k,m) by switch 404, controlled by the index of theminimum sum of squared distances; that is Ŷ_(k)=Ŷ_(k,m=min) for whiche_(k,m=min) is the minimum value of e_(k,m).

The computational complexity of the blind phase search method describedabove depends on the required number of test phase angles; this numbercan be high for high-order M-QAM due to the high requirement on thephase resolution. Since testing each phase angle requires the series ofdigital operations shown in FIG. 4A, the overall number digitaloperations shown in FIG. 4B is extremely computationally intensive forhigh-order M-QAM.

FIG. 5A shows a schematic block diagram for a two-stage carrier phaserecovery algorithm according to an embodiment. The received signal X_(k)401 is first inputted into coarse blind phase search block 502. Theoperations in this block are similar to those discussed above withrespect to FIG. 4A and FIG. 4B. In this instance, however, only a subset(less than M) of the possible carrier phases are tested. In anembodiment, B carrier phases, where B<M/2, are tested. The output ofcoarse blind phase search block 502 is a rough estimate of the optimalphase angle. The output of coarse blind phase search block 502 is output501, which is the decoded symbol Ŷ_(k) ⁽¹⁾. The quantity Ŷ_(k) ⁽¹⁾ andthe original signal X_(k) are then inputted into maximum likelihoodphase recovery and decoding block (MLPRD) 504. The output of MLPRD block504 is output 507, denoted Ŷ_(k) ⁽²⁾, which is a refined estimate.

The operations within MLPRD block 504 proceeds as follows. The quantityŶ_(k) ⁽¹⁾ and the original signal X_(k) inputted into maximum likelihood(ML) calculation block 506, in which the following calculations areperformed [J. G. Proakis, Digital Communications, 4^(th) edition,Chapter 6, pg. 348, McGraw-Hill (2000)]:

$\begin{matrix}{{H_{k} = {\sum\limits_{n = {k - N + 1}}^{k + N}{X_{n}\left\lbrack {\hat{Y}}_{n}^{(1)} \right\rbrack}^{*}}}{\phi_{k}^{ML} = {{\tan^{- 1}\left( {{{Im}\left\lbrack H_{k} \right\rbrack}/{{Re}\left\lbrack H_{k} \right\rbrack}} \right)}.}}} & ({E4})\end{matrix}$Here φ_(k) ^(ML) is the refined phase estimate.

The output of maximum likelihood calculation block 506 is output 503,which is the phase rotation factor e^(−jφ) ^(ML) . The phase rotationfactor and the signal X_(k) are inputted into multiplier 508. The outputof multiplier 508 is output 505, which is the rotated symbolX_(k)e^(−jφ) ^(ML) . The rotated symbol is inputted into decision block510, which makes the best estimate of the originally transmitted signalbased on the received signal. The output of decision block 510 is output507, which is the refined estimate Ŷ_(k) ⁽²⁾.

The estimate Ŷ_(k) ⁽²⁾ can be further refined by a second stage ofmaximum likelihood phase recovery and decoding. FIG. 5B shows an S-stagealgorithm with an initial coarse blind phase search 502 stage followedby S-1 MLPRD stages. Shown in FIG. 5B are MLPRD-1 504 second stage,MPLRD-2 524 third stage, and MLPRD-(S-1) 554 S-th stage. The output ofMPLRD-2 524 is output 509, which is Ŷ_(k) ⁽³⁾. The last stage receivesoutput 551 (Ŷ_(k) ^((S-1))) from the previous stage and outputs output553, which is the final value (Ŷ_(k) ^(S)).

Since only a rough location of the optimal carrier phase needs to beestimated in coarse blind phase search stage 502, the required number oftest phase angles can be reduced substantially from the required numberof test phase angles used in the single-stage blind phase search method400 (FIG. 4A and FIG. 4B). In practice, the required number of digitaloperations for a ML-phase estimate (E4) is equivalent to testing onlyone phase angle using the blind phase search method. As a consequence,the required computational efforts for the multi-stage carrier recoveryalgorithm can be substantially lower than that required by thesingle-stage blind phase search method.

FIG. 6 shows the results of numerical simulation for square 64-QAM. Thehorizontal axis 602 represents the log₂ of the required effective numberof test phase angles. The vertical axis 604 represents the bit errorrate (BER). Plot 604 shows the results for the prior-art single-stageblind phase search method; plot 606 shows the results for a two-stagemethod (initial coarse blind phase search stage followed by one MPLRDstage) according to an embodiment; and plot 608 shows the results for athree-stage method (initial coarse blind phase search stage followed bytwo MPLRD stages) according to an embodiment. For these simulations, thefollowing assumptions were used: (1) The baud rate is equal to 38symbols/s, corresponding to 224 Gbits/s; (2) The received opticalsignal-to-noise ratio (OSNR) at 0.1 nm noise bandwidth is equal to 28dB; (3) The 3-dB receiver electrical bandwidth is equal to 0.55×baudrate; (4) The laser linewidth for both the signal source and the localoscillator is equal to 100 kHz; (5) The signal entering into the carrierphase recovery block is sampled at one sample per symbol and the usedblock/filter width for phase recovery is assumed to be 28, which isclose to the optimal value; and (6) A cascaded multi-modulus algorithmbased adaptive equalizer has been employed prior to the carrier phaserecovery block to equalize the receiver filtering effects.

As seen in the plots in FIG. 6, to achieve the optimal performance, thesingle-stage blind phase search method needs to test about 64 differentphase angles, while the three-stage algorithm only needs to equivalentlytest 18 different phase angles, resulting in a reduction ofcomputational efforts by a factor more than 3 (the two-stage algorithmrequires 20 equivalent test phase angles). Because the multi-stagemethod employs a feed-forward configuration and involves all the currentsymbols for the phase estimate, it can achieve the same linewidthtolerance as the single-stage blind phase search method, but withsignificantly reduced computational power.

FIG. 7 shows a flowchart of a multi-stage carrier phase recovery method,according to an embodiment. In step 702, a digital signal X_(k) isreceived. The process then passes to step 704, in which an initialcoarse blind phase search stage is performed to generate Ŷ_(k) ⁽¹⁾, thefirst estimate of the decoded symbol. The process then passes to step706, in which a first maximum likelihood phase recovery and decodingstage is performed to generate Ŷ_(k) ⁽²⁾, the second (refined) estimateof the decoded symbol. The process then passes to step 708, in which asecond maximum likelihood phase recovery and decoding stage is performedto generate Ŷ_(k) ⁽³⁾, the third (further refined) estimate of thedecoded symbol.

The multi-stage carrier phase recovery method described above wasillustrated with two-stages and three-stages for a M-QAM signal. Ingeneral, the multi-stage carrier phase recovery method can includeN-stages, where N is an integer greater than or equal to two. Ingeneral, the multi-stage carrier phase recovery method can be used withany modulation technique.

The multi-stage carrier phase recovery method described above used acoarse blind phase search as the initial coarse phase recovery stage. Ingeneral, other phase estimate methods can be used for the first stage.As one example, the carrier phase can be initially estimated by thetraditional decision-directed phase-locked loop, which is widely usedfor carrier recovery in RF communications (A. Tarighat, “DigitalAdaptive Phase Noise Reduction in Coherent Optical Links”, J. LigtwaveTechnology, Vol. 24, No. 3, March 2006, pp. 1269-1276 and I. Fatadin,“Compensation of Frequency Offset for Differentially Encoded 16- and64-QAM in the Presence of Laser Phase Noise”, IEEE Photonics TechnologyLetters, Vol. 22, No. 3, Feb. 1, 2010, pp. 176-178). The decided signalfollowing this first-stage carrier phase recovery can then be used asthe reference signal for the second-stage carrier phase recovery. Asdiscussed above, however, a stand-alone decision-directed phase-lockedloop has poor laser phase noise tolerance. In embodiments of themulti-stage carrier phase recovery method, a decision-directedphase-locked loop is used only in the first stage for coarse phaserecovery, and carrier phase estimate accuracy can be improved by thefollowing carrier phase recovery stages.

In general, the first stage is referred to herein as coarse phaserecovery. Coarse phase recovery refers to any phase recovery scheme thatcan recover the carrier phase to some extent such that the resultingdecision error is smaller than the case without applying such phaserecovery scheme (but not close to the optimum). For example, assume thatthe measured decision error rate without using any phase recovery schemeis A, and the measured decision error rate by introducing one specificphase recovery scheme is B. As long as B<A, then the introduced phaserecovery scheme is a coarse phase recovery scheme. Therefore,embodiments of coarse phase recovery include a coarse blind phase searchmethod, a decision-directed phase-locked loop, and other phase recoveryschemes. In general, B is not close to the optimum value (as determinedby user-specified criteria), since the coarse phase recovery schemeemphasizes reduced computational complexity and increased computationalspeed rather than high accuracy.

The multi-stage carrier phase recovery method described above used amaximum likelihood phase estimate in the second and higher stages. Ingeneral, other phase estimate methods can be used for the second andhigher stages. As one example, the carrier phase can be estimated bydirectly calculating the average phase rotation of the original receivedsignal (undecoded signal prior to carrier phase recovery) relative tothe decoded signal obtained from the previous stage; this method isreferred to herein as an average phase rotation estimate. As anotherexample, the blind phase search method with a refined (reduced orrestricted) phase scan range can be used in the second and higherstages; this method is referred to herein as a restricted blind phasesearch estimate. Note that different methods be used for differentstages; for example, a maximum likelihood estimate can be used for thesecond stage, and a restricted blind phase search estimate can be usedfor the third stage.

Since the multi-stage carrier phase recovery method is performed afteroptical to electronic conversion (see FIG. 1), it is also applicable toother carriers. For example, a radio-frequency (RF) carrier can be mixedin a RF coherent mixer with a reference RF signal generated by a localRF oscillator. The output of the RF coherent mixer can then be convertedto different electrical signals and digitized.

FIG. 8 shows an example of a computational system 802 for performing amulti-stage carrier phase recovery process. One skilled in the art canconstruct the computational system 802 from various combinations ofhardware and software (including firmware). One skilled in the art canconstruct the computational system 802 from various combinations ofelectronic components, such as general purpose microprocessors, digitalsignal processors (DSPs), application-specific integrated circuits(ASICs), field-programmable gate arrays (FPGAs), random access memory,and non-volatile read-only memory.

Computational system 802 comprises computer 804, which includes adigital signal processor (DSP) 806, memory 808, and data storage device810. Data storage device 810 comprises at least one non-transitory,persistent, tangible computer readable medium, such as non-volatilesemiconductor memory (data storage device 810 can also comprise othernon-transitory, persistent, tangible computer readable medium withsufficiently high data transfer rates).

Computational system 802 further comprises input/output interface 820,which interfaces computer 804 with input/output device 840. Data,including computer executable code can be transferred to and fromcomputer 804 via input/output interface 820. Computational system 802further comprises digital signal interface A 822, which interfacescomputer 804 with digital signal source 842. An example of digitalsignal source 842 is a DSP that transmits digital signal X_(k).Computational system 802 further comprises digital signal interface B824, which interfaces computer 804 with digital signal receiver 844. Anexample of digital signal receiver 844 is a DSP that receives decodedsymbol Ŷ_(k) ⁽³⁾.

As is well known, a computer operates under control of computersoftware, which defines the overall operation of the computer andapplications. DSP 806 controls the overall operation of the computer andapplications by executing computer program instructions that define theoverall operation and applications. The computer program instructionscan be stored in data storage device 810 and loaded into memory 808 whenexecution of the program instructions is desired. The method steps shownin the flowchart in FIG. 7 can be defined by computer programinstructions stored in memory 808 or in data storage device 810 (or in acombination of memory 808 and data storage device 810) and controlled bythe DSP 806 executing the computer program instructions. For example,the computer program instructions can be implemented as computerexecutable code programmed by one skilled in the art to performalgorithms implementing the method steps shown in the flowchart in FIG.7. Accordingly, by executing the computer program instructions, the DSP806 executes algorithms implementing the method steps shown in theflowchart in FIG. 7.

The foregoing Detailed Description is to be understood as being in everyrespect illustrative and exemplary, but not restrictive, and the scopeof the inventive concept disclosed herein is not to be determined fromthe Detailed Description, but rather from the claims as interpretedaccording to the full breadth permitted by the patent laws. It is to beunderstood that the embodiments shown and described herein are onlyillustrative of the principles of the present disclosure and thatvarious modifications may be implemented by those skilled in the artwithout departing from the scope and spirit of the disclosure. Thoseskilled in the art could implement various other feature combinationswithout departing from the scope and spirit of the disclosure.

The invention claimed is:
 1. A method for carrier phase recovery of acarrier wave modulated with a plurality of information symbols, themethod comprising: receiving a digital signal corresponding to aninformation symbol; generating a first estimate of an optimal phaseangle corresponding to the digital signal by performing a coarse blindphase search; and generating a second estimate of the optimal phaseangle corresponding to the digital signal by inputting the digitalsignal and the first estimate of the optimal phase angle into a firstmaximum likelihood phase recovery and decoding block.
 2. The method ofclaim 1, further comprising: generating a third estimate of the optimalphase angle corresponding to the digital signal by inputting the digitalsignal and the second estimate of the optimal phase angle into a secondmaximum likelihood phase recovery and decoding block.
 3. The method ofclaim 1, wherein the generating a second estimate of the optimal phaseangle precedes the generating a third estimate of the optimal phaseangle.
 4. The method of claim 1, wherein generating the second estimateof the optimal phase angle is a part of a coarse phase recovery.
 5. Themethod of claim 1, wherein the information symbol represents data. 6.The method of claim 1, wherein the carrier wave is modulated with M-aryquadrature amplitude modulation and the performing the coarse blindphase search comprises testing fewer than M test phase angles.
 7. Themethod of claim 1, wherein the carrier wave is an optical carrier wave.8. The method of claim 2, further comprising: generating a fourthestimate of the optimal phase angle corresponding to the digital signalby inputting the digital signal and the third estimate of the optimalphase angle into a third maximum likelihood phase recovery and decodingblock.
 9. The method of claim 8, wherein the generating the thirdestimate of the optimal phase angle precedes the generating the fourthestimate of the optimal phase angle k.
 10. An apparatus for carrierphase recovery of a carrier wave modulated with a plurality ofinformation symbols, the apparatus comprising: a processor; and a memoryto store computer program instructions, the computer programinstructions, when executed on the processor, cause the processor toperform operations comprising: receiving a digital signal correspondingto an information symbol, wherein the information symbol representsdata; generating a first estimate of an optimal phase anglecorresponding to the digital signal by performing a coarse blind phasesearch; and generating a second estimate of the optimal phase anglecorresponding to the digital signal by inputting the digital signal andthe first estimate of the optimal phase angle into a first maximumlikelihood phase recovery and decoding block.
 11. The apparatus of claim10, the operations further comprising: generating a third estimate ofthe optimal phase angle corresponding to the digital signal by inputtingthe digital signal and the second estimate of the optimal phase angleinto a second maximum likelihood phase recovery and decoding block. 12.The apparatus of claim 10, wherein the generating a second estimate ofthe optimal phase angle precedes the generating a third estimate of theoptimal phase angle.
 13. The apparatus of claim 10, wherein generatingthe second estimate of the optimal phase angle is a part of a coarsephase recovery.
 14. The apparatus of claim 11, the operations furthercomprising: generating a fourth estimate of the optimal phase anglecorresponding to the digital signal by inputting the digital signal andthe third estimate of the optimal phase angle into a third maximumlikelihood phase recovery and decoding block.
 15. The apparatus of claim14, wherein the generating the third estimate of the optimal phase angleprecedes the generating the fourth estimate of the optimal phase anglek.
 16. A non-transitory computer readable medium storing computerprogram instructions for carrier phase recovery of a carrier wavemodulated with a plurality of information symbols, which, when executedon a processor, cause the processor to perform operations comprising:receiving a digital signal corresponding to an information symbol,wherein the information symbol represents data; generating a firstestimate of an optimal phase angle corresponding to the digital signalby performing a coarse blind phase search; generating a second estimateof the optimal phase angle corresponding to the digital signal byinputting the digital signal and the first estimate of the optimal phaseangle into a first maximum likelihood phase recovery and decoding block.17. The non-transitory computer readable medium of claim 16, theoperations further comprising: generating a third estimate of theoptimal phase angle corresponding to the digital signal by inputting thedigital signal and the second estimate of the optimal phase angle into asecond maximum likelihood phase recovery and decoding block.
 18. Thenon-transitory computer readable medium of claim 16, wherein thegenerating a second estimate of the optimal phase angle precedes thegenerating a third estimate of the optimal phase angle.
 19. Thenon-transitory computer readable medium of claim 17, the operationsfurther comprising: generating a fourth estimate of the optimal phaseangle corresponding to the digital signal by inputting the digitalsignal and the third estimate of the optimal phase angle into a thirdmaximum likelihood phase recovery and decoding block.
 20. Thenon-transitory computer readable medium of claim 16, wherein thegenerating the third estimate of the optimal phase angle precedes thegenerating the fourth estimate of the optimal phase angle.